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On a Certain Inequality
Raitis Ozols, University of Latvia, [email protected]
Elementary algebra forms the basis for mathematical analysis and calculus.
Inequalities are especially important here. A lot of them were invented and proved as
convenient tools for investigating the classical problems of continuous mathematics. On theother side, inequalities themselves can provide stimulus for serious research already on high-
school level. John Littlewood has said in his famous book [1] that some classical inequalities
still today could turn into new important theorems if given to great mathematician in the
moment of inspiration. So there is no wonder that most famous of them have a lot of proofs;
e.g., the inequality between arithmetic and geometric means has 50 of them (see [2]). The
Cauchy-Schwartz inequality is an another example (see [3]).
The methods of inequality proving are also of the main topics in the preparation of
students to mathematical competitions. There are many teaching aids developed for this. The
main methods usually considered are as follows:
a) identical transformations (often to the form 02 kx )b) strengthening of the inequality,c) mathematical induction,d) derivative and integral,e) interpretations,f) applications of other inequalities (Cauchy, Cauchy-Schwartz, Jensen, etc.).
Many books on special methods have been published, e.g., [4].
In this paper we consider a proof of a double inequality L(a, b) M0,5(a, b) P(a, b) for
positive a and b. Hereab
abbaL
lnln),(
= (ab) and L(a, a) = a is the logarithmic mean,
2
5,0
2
),(
+=
babaM is the square root mean and
=
ab
abbaP
/arctan4
),( (ab) and
P(a, a) = a is the Seiffert mean. The proof can be of interest because it starts from the obvious
inequality 22 )1(0)1(